Runtime_Complexity_Innermost_Rewriting 2019-04-01 06.40 pair #433313788

details

property	value
status	complete
benchmark	fold.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n128.star.cs.uiowa.edu
space	Frederiksen_Glenstrup

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	5.00782 seconds
cpu usage	15.8054
user time	14.5243
system time	1.28109
max virtual memory	1.9315588E7
max residence set size	3852996.0

stage attributes

key	value
starexec-result	WORST_CASE(Omega(n^1), O(n^1))

output

15.38/4.86	WORST_CASE(Omega(n^1), O(n^1))
15.71/4.87	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
15.71/4.87	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
15.71/4.87	
15.71/4.87	
15.71/4.87	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
15.71/4.87	
15.71/4.87	(0) CpxTRS
15.71/4.87	(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
15.71/4.87	(2) CdtProblem
15.71/4.87	    (3) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
15.71/4.87	    (4) CdtProblem
15.71/4.87	    (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
15.71/4.87	    (6) CdtProblem
15.71/4.87	    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
15.71/4.87	    (8) CdtProblem
15.71/4.87	    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 76 ms]
15.71/4.87	    (10) CdtProblem
15.71/4.87	    (11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
15.71/4.87	    (12) BOUNDS(1, 1)
15.71/4.87	(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
15.71/4.87	(14) TRS for Loop Detection
15.71/4.87	    (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
15.71/4.87	    (16) BEST
15.71/4.87	        (17) proven lower bound
15.71/4.87	            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
15.71/4.87	            (19) BOUNDS(n^1, INF)
15.71/4.87	        (20) TRS for Loop Detection
15.71/4.87	
15.71/4.87	
15.71/4.87	----------------------------------------
15.71/4.87	
15.71/4.87	(0)
15.71/4.87	Obligation:
15.71/4.87	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
15.71/4.87	
15.71/4.87	
15.71/4.87	The TRS R consists of the following rules:
15.71/4.87	
15.71/4.87	   foldl(x, Cons(S(0), xs)) -> foldl(S(x), xs)
15.71/4.87	   foldl(S(0), Cons(x, xs)) -> foldl(S(x), xs)
15.71/4.87	   foldr(a, Cons(x, xs)) -> op(x, foldr(a, xs))
15.71/4.87	   foldr(a, Nil) -> a
15.71/4.87	   foldl(a, Nil) -> a
15.71/4.87	   notEmpty(Cons(x, xs)) -> True
15.71/4.87	   notEmpty(Nil) -> False
15.71/4.87	   op(x, S(0)) -> S(x)
15.71/4.87	   op(S(0), y) -> S(y)
15.71/4.87	   fold(a, xs) -> Cons(foldl(a, xs), Cons(foldr(a, xs), Nil))
15.71/4.87	
15.71/4.87	S is empty.
15.71/4.87	Rewrite Strategy: INNERMOST
15.71/4.87	----------------------------------------
15.71/4.87	
15.71/4.87	(1) CpxTrsToCdtProof (UPPER BOUND(ID))
15.71/4.87	Converted Cpx (relative) TRS to CDT
15.71/4.87	----------------------------------------
15.71/4.87	
15.71/4.87	(2)
15.71/4.87	Obligation:
15.71/4.87	Complexity Dependency Tuples Problem
15.71/4.87	
15.71/4.87	Rules:
15.71/4.87	   foldl(z0, Cons(S(0), z1)) -> foldl(S(z0), z1)
15.71/4.87	   foldl(S(0), Cons(z0, z1)) -> foldl(S(z0), z1)
15.71/4.87	   foldl(z0, Nil) -> z0
15.71/4.87	   foldr(z0, Cons(z1, z2)) -> op(z1, foldr(z0, z2))
15.71/4.87	   foldr(z0, Nil) -> z0
15.71/4.87	   notEmpty(Cons(z0, z1)) -> True
15.71/4.87	   notEmpty(Nil) -> False
15.71/4.87	   op(z0, S(0)) -> S(z0)
15.71/4.87	   op(S(0), z0) -> S(z0)
15.71/4.87	   fold(z0, z1) -> Cons(foldl(z0, z1), Cons(foldr(z0, z1), Nil))
15.71/4.87	Tuples:
15.71/4.87	   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
15.71/4.87	   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
15.71/4.87	   FOLDL(z0, Nil) -> c2
15.71/4.87	   FOLDR(z0, Cons(z1, z2)) -> c3(OP(z1, foldr(z0, z2)), FOLDR(z0, z2))
15.71/4.87	   FOLDR(z0, Nil) -> c4
15.71/4.87	   NOTEMPTY(Cons(z0, z1)) -> c5
15.71/4.87	   NOTEMPTY(Nil) -> c6
15.71/4.87	   OP(z0, S(0)) -> c7
15.71/4.87	   OP(S(0), z0) -> c8
15.71/4.87	   FOLD(z0, z1) -> c9(FOLDL(z0, z1), FOLDR(z0, z1))
15.71/4.87	S tuples:
15.71/4.87	   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
15.71/4.87	   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
15.71/4.87	   FOLDL(z0, Nil) -> c2
15.71/4.87	   FOLDR(z0, Cons(z1, z2)) -> c3(OP(z1, foldr(z0, z2)), FOLDR(z0, z2))
15.71/4.87	   FOLDR(z0, Nil) -> c4
15.71/4.87	   NOTEMPTY(Cons(z0, z1)) -> c5
15.71/4.87	   NOTEMPTY(Nil) -> c6
15.71/4.87	   OP(z0, S(0)) -> c7
15.71/4.87	   OP(S(0), z0) -> c8
15.71/4.87	   FOLD(z0, z1) -> c9(FOLDL(z0, z1), FOLDR(z0, z1))
15.71/4.87	K tuples:none
15.71/4.87	Defined Rule Symbols:   foldl_2, foldr_2, notEmpty_1, op_2, fold_2
15.71/4.87	
15.71/4.87	Defined Pair Symbols:   FOLDL_2, FOLDR_2, NOTEMPTY_1, OP_2, FOLD_2
15.71/4.87

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actions all output return to Runtime_Complexity_Innermost_Rewriting 2019-04-01 06.40